As for the grudge between Whitebeard Jr. and Zephyr, the World Government was even somewhat pleased to see it happen.
Doflamingo, the head of the Shichibukai, had grown far too close to the Navy. Among the remaining Shichibukai, Sand Crocodile and Gecko Moria had long since been controlled by Doflamingo. Kuma belonged to the Revolutionary Army and could be considered Doflamingo's ally. Even Hawkeye, who had always been a loner, had exchanged swordsmanship with Doflamingo on several occasions and had even owed him a favor.
Perhaps the World Government had no way of detecting the connections between them, but it could still sense that the other Shichibukai seemed to follow Doflamingo's lead. At the very least, Doflamingo carried considerable weight among the Shichibukai.
Under such circumstances, the World Government naturally did not want any further contact between the Shichibukai and the Navy. The World Government had established the Shichibukai to balance the situation across the seas. That balance was not merely aimed at pirates—it applied to the Navy as well!
Otherwise, the World Government could simply have sought out powerful individuals with clean records and no notorious past to serve as Shichibukai. The sea was vast, and there had never been a shortage of strong people who belonged neither to the pirates nor the Navy. Otherwise, a single World Military Draft would never have unearthed two Admiral-level talents for the Navy.
The reason pirates were always chosen as Shichibukai was obvious: the World Government did not want them drawing closer to the Navy.
Of course, the current Shichibukai policy had already left the World Government in a difficult position—if it continued maintaining the system, Doflamingo could keep clinging to the World Government and draining it dry. But if it abolished the Shichibukai, Doflamingo would immediately turn hostile toward the World